When does Vader first learn about Luke?

My guess would be that he learns about Luke in the timespan between ANH and ESB. I think the official cannon timeline has those two movies 2-3 years apart. So it would be easy enough as they are chasing them across the galaxy to capture a few people, interrogate them and find out the name of the pilot who blew up the Death Star.

He doesn't learn about Leia until he and Luke are in the throne room on the new Death Star in ROTJ - Luke thinks about her and Vader reads his thoughts.

My question (totally off topic but came to mind with the timespan mentioned above) is how long does it take to build a Death Star? We see Death Star 1 in the early stage of construction at the end of Ep 3, and it seems to be a fairly new weapon when we are introduced to it in Ep 4 - so a good 15 years or so to build it. But then we see DS1 destroyed and DS2 is well along in the construction phase in just about a 5 year span from Ep 4 to Ep 6.

Did the Empire learn so much about building it the first time that they were able to cut construction time by 2/3s with the second one, or did they use hundreds or thousands of labor conscripts to build it faster?

It's important to note that Lucas was making and changing things on the fly, so for your own enjoyment, it helps to not overthink it.